In addition, \(\mathfrak {A}\)  defined by Eq.(REF ) is not continuous across the bounce, as the coefficient \(2V_{,\phi }(\phi )\sqrt{r}\)  in general does not vanish at the bounce.
In [1]}, [2]}, [3]}
\(\mathfrak {A}_{-}\)  appearing in Eq.(REF ) was replaced by \(U(\phi ) (\equiv \Omega ^2_{+})\)  over the whole process of the evolution of the homogeneous universe,
where
\(\Omega ^2_{\pm } \equiv a^2\left[\mathfrak {f}^2 V(\phi ) \pm 2 \mathfrak {f} V_{,\phi }(\phi ) + V_{,\phi \phi }(\phi )\right],\) 
